As far as the author knows, the paper by Linares and Ponce [1]} was the first to establish a unique continuation result for the solutions of the CH equation. Soon after, the author of the present work applied their ideas to the DGH equation [2]} and went further, connecting them with conserved quantities. These two ingredients not only give a demonstration for unique continuation of the DGH equation, but also provides a machinery to prove investigate the non-existence of non-trivial compactly supported solutions of the DGH equation and, as a particular case, of the CH equation. In the present paper we make an in-depth investigation on the ideas introduced in [1]}, [2]}, which were better explored in the Section .
